6 RICE 



By Taylor's theorem this is equal to 



ART. B 



^X'"^^'^^ 



r = 1 s = 1 



d^rbg. 



hqrhq^ + etc. 



where we write f{q) briefly for fiqi, q^, .... g„). 

 This difference we now write in the form 



+ 



2! 



ss 



mq) 



1 



:, . . ^.^a[4-etc. (2) 



As is reduced in value, the numerical magnitude of the term 

 in preponderates more and more over the terms in 6'^, 9^, .... 

 (apart from discontinuities arising in the differential coefficients, 

 a state of affairs which we cannot discuss here). The sign of 

 this term will therefore determine whether f(q + dq) is greater 

 or less than/(5). If /(g + 5g) is greater than /(g) for any values 

 of {qr + 8qr) consistent with the condition imposed, it is neces- 

 sary that 



^ 5/(g) 

 ^qr 



r = 1 



^r 







(3) 



for apy possible sets of values of (Ir)^ since if the expression on 

 the left-hand side of (3) were positive for a set of values of (|r), 

 it would be negative for the set with opposite signs, and so 

 f(q + 8q) would not be greater than f(q) for all possible sets of 

 (qr + 8qr). If the quantities (^r) were perfectly arbitrary this 

 would necessitate the n conditions, c>f(q)/dqr = 0. However, 

 they are not arbitrary; for by (1) they satisfy the condition 



S 



c>4>(q), , e 



ss 



c)V(g) 



.=1 ^Qr "" ■ 2!(^^jfr<bg.dg. 

 For very small values of 6, this becomes 



^r ?s 



+ etc. = 0. 



yA 5</>(g) 



^1 ^^r 



^r = 0. 



(4) 



